A box with a square base and no top is to be made from a square piece of carboard by cutting 4 in . squares from each corner and folding up the sides. The box is to hold \( 676 \mathrm{in}^{3} \). How big a piece of cardboard is needed? Your answer is: \( \square \mathrm{in} \). by.

To find the size of the piece of cardboard needed to make the box, we need to determine the dimensions of the box's base and height. Given: - The box has a square base and no top. - 4 in. squares are cut from each corner and folded up to form the sides. - The box is to hold \( 676 \mathrm{in}^{3} \). Let's denote the side length of the square base as \( x \) inches. When the 4 in. squares are cut from each corner, the new side length of the base becomes \( x - 8 \) inches. The height of the box is the length of the side of the square cut from each corner, which is 4 inches. The volume of the box is given by the formula: \[ \text{Volume} = \text{Base Area} \times \text{Height} \] Substitute the given values: \[ 676 = (x - 8)^2 \times 4 \] Now, we can solve for \( x \) to find the size of the piece of cardboard needed. Solve the quadratic equation by following steps: - step0: Solve by factoring: \(676=\left(x-8\right)^{2}\times 4\) - step1: Reorder the terms: \(676=4\left(x-8\right)^{2}\) - step2: Swap the sides: \(4\left(x-8\right)^{2}=676\) - step3: Expand the expression: \(4x^{2}-64x+256=676\) - step4: Move the expression to the left side: \(4x^{2}-64x-420=0\) - step5: Factor the expression: \(4\left(x-21\right)\left(x+5\right)=0\) - step6: Divide the terms: \(\left(x-21\right)\left(x+5\right)=0\) - step7: Separate into possible cases: \(\begin{align}&x-21=0\\&x+5=0\end{align}\) - step8: Solve the equation: \(\begin{align}&x=21\\&x=-5\end{align}\) - step9: Rewrite: \(x_{1}=-5,x_{2}=21\) The solutions for \( x \) are \( x = -5 \) and \( x = 21 \). Since the side length of the square base cannot be negative, the size of the piece of cardboard needed is \( 21 \) inches.